Hostname: page-component-74656f9884-rdgjk Total loading time: 0 Render date: 2026-11-07T21:44:33.406Z Has data issue: false hasContentIssue false
  • English
  • Français

A certain structure of Artin groups and the isomorphism conjecture

Part of: Topological manifolds Whitehead groups and $K_1$ $K$-theory of forms Special aspects of infinite or finite groups Differential topology

Published online by Cambridge University Press:  21 May 2020

S.K. Roushon
S.K. Roushon*
Affiliation:
School of Mathematics, Tata Institute, Homi Bhabha Road, Mumbai 400005, India URL: http://www.math.tifr.res.in/~roushon
*
e-mail: roushon@math.tifr.res.in
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We observe an inductive structure in a large class of Artin groups of finite real, complex and affine types and exploit this information to deduce the Farrell–Jones isomorphism conjecture for these groups.

Keywords

Artin groupisomorphism conjectureWhitehead groupreduced projective class groupsurgery obstruction groupWaldhausen A-theory

MSC classification

Primary: 19B99: None of the above, but in this section
Secondary: 19G24: $L$-theory of group rings 20F36: Braid groups; Artin groups 57R67: Surgery obstructions, Wall groups 57N37: Isotopy and pseudo-isotopy

Information

Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 4 , August 2021 , pp. 1153 - 1170
Check for updates
Copyright
© Canadian Mathematical Society 2020

References

Allcock, D., Braid pictures of Artin groups. Trans. Amer. Math. Soc. 354(2002), no. 9, 3455–3474.CrossRefGoogle Scholar
Bartels, A. and Bestvina, M., The Farrell-Jones Conjecture for mapping class groups. Invent. Math. 215(2019), no. 2, 651–712.CrossRefGoogle Scholar
Bartels, A. and Lück, W., The Borel conjecture for hyperbolic and $CAT(0)$ groups . Ann. Math. 175(2012), no. 2, 631–689.CrossRefGoogle Scholar
Brieskorn, E., Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe. Invent. Math. 12(1971), 57–61.CrossRefGoogle Scholar
Brieskorn, E., Sur les groupes de tresses [d’après V.I. Arnol’d]. In: Séminaire Bourbaki 24ème année (1971/1972), Exp. No. 401, Lecture Notes in Math., 317, Springer, Berlin, 1973, pp. 21–44.Google Scholar
Charney, R., Problems related to Artin groups. 2006. http://people.brandeis.edu/~charney/papers/Artin_probs.pdf Google Scholar
Corran, R., Lee, E., and Lee, S., Braid groups of imprimitive complex reflection groups. J. Algebra 427(2015), 387–425.CrossRefGoogle Scholar
Coxeter, H. S. M., The complete enumeration of finite groups of the form ${R}_i^2={\left({R}_i{R}_j\right)}^{k_{ij}}=1$ . J. Lond. Math. Soc. 10(1935), 21–25.CrossRefGoogle Scholar
Deligne, P., Les immeubles des groupes de tresses généralisés . Invent. Math. 17(1972), 273–302.CrossRefGoogle Scholar
Fadell, E. and Neuwirth, L., Configuration spaces. Math. Scand. 10(1962), 111–118.CrossRefGoogle Scholar
Farrell, F. T. and Jones, L. E., Isomorphism conjectures in algebraic $K$ -theory . J. Amer. Math. Soc. 6(1993), 249–297.Google Scholar
Farrell, F. T. and Roushon, S. K., The Whitehead groups of braid groups vanish. In. Math. Res. Notices 10(2000), 515–526.CrossRefGoogle Scholar
Garside, F. A., The braid group and other groups. Quart. J. Math. Oxford 20(1969), 235–254.CrossRefGoogle Scholar
Hambleton, I. and Taylor, L. R., A guide to the calculation of surgery obstruction groups for finite groups. Surveys on surgery theory, 1, Princeton University Press, Princeton, NJ, 2000, pp. 225–274.Google Scholar
Hsiang, W.-C., Geometric applications of algebraic $K$ -theory . Proc. Int. Cong. Math. 1(1983), 99–118.Google Scholar
Humphreys, J. E., Reflection groups and Coxeter groups . Cambridge Studies in Advanced Mathematics, 29, Cambridge University Press, Cambridge, UK, 1990.Google Scholar
Lück, W., Assembly maps. In: Miller, H. R. (ed.), Handbook of homotopy theory, CRC Press/Chapman & Hall, Assembly Maps, Boca Raton, FL, 2019, pp. 853–892.Google Scholar
Orlik, P. and Terao, H., Arrangements of hyperplanes . Springer-Verlag, Berlin, 1992.CrossRefGoogle Scholar
Paolini, G. and Salvetti, M., Proof of the $K\left(\pi, 1\right)$ conjecture for affine Artin groups. Preprint, 2019. arXiv:1907.11795.CrossRefGoogle Scholar
Roushon, S. K., The Farrell-Jones isomorphism conjecture for $3$ -manifold groups . J. K-Theory 1(2008), 49–82.CrossRefGoogle Scholar
Roushon, S. K., The isomorphism conjecture for $3$ -manifold groups and $K$ -theory of virtually poly-surface groups . J. K-Theory 1(2008), 83–93.CrossRefGoogle Scholar
Roushon, S. K., Surgery groups of the fundamental groups of hyperplane arrangement complements . Arch. Math. (Basel) 96(2011) no. 5, 491–500.CrossRefGoogle Scholar
Roushon, S. K., The isomorphism conjecture for groups with generalized free product structure . Handbook of Group Actions, II, Higher Education Press and International Press, Beijing, 2014, pp. 77–119.Google Scholar
Shaneson, J., Wall’s surgery obstruction groups for $G\times \mathbb{Z}$ . Ann. Math. 90(1969), no. 2, 296–334.CrossRefGoogle Scholar
Shephard, G. C. and Todd, J. A., Finite unitary reflection groups . Canad. J. Math. 25(1954), 274–304.CrossRefGoogle Scholar
Thurston, W. P., Three-dimensional geometry and topology . Mathematical Sciences Research Institute Notes, Berkeley, CA, 1991.Google Scholar
Wegner, C., The $\textit{K}$ -theoretic Farrell-Jones conjecture for ${CAT}(0)$ -groups. Proc. Amer. Math. Soc. 140(2012), 779–793.CrossRefGoogle Scholar